Optimal. Leaf size=174 \[ \frac{8 a^2 (63 A+47 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (63 A+22 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{21 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.476666, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4089, 4010, 4001, 3793, 3792} \[ \frac{8 a^2 (63 A+47 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (63 A+22 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{21 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4089
Rule 4010
Rule 4001
Rule 3793
Rule 3792
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac{2 \int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{1}{2} a (9 A+4 C)+\frac{3}{2} a C \sec (c+d x)\right ) \, dx}{9 a}\\ &=\frac{2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac{4 \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{15 a^2 C}{4}+\frac{1}{4} a^2 (63 A+22 C) \sec (c+d x)\right ) \, dx}{63 a^2}\\ &=\frac{2 (63 A+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac{2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac{1}{105} (63 A+47 C) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac{2 a (63 A+47 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac{2 (63 A+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac{2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}+\frac{1}{315} (4 a (63 A+47 C)) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{8 a^2 (63 A+47 C) \tan (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (63 A+47 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac{2 (63 A+22 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{315 d}+\frac{2 C \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{9 d}+\frac{2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{21 a d}\\ \end{align*}
Mathematica [A] time = 1.24025, size = 121, normalized size = 0.7 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} ((567 A+748 C) \cos (c+d x)+(882 A+748 C) \cos (2 (c+d x))+189 A \cos (3 (c+d x))+189 A \cos (4 (c+d x))+693 A+136 C \cos (3 (c+d x))+136 C \cos (4 (c+d x))+752 C)}{630 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.29, size = 130, normalized size = 0.8 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 378\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+272\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+189\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+136\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+63\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+102\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+85\,C\cos \left ( dx+c \right ) +35\,C \right ) }{315\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.503773, size = 319, normalized size = 1.83 \begin{align*} \frac{2 \,{\left (2 \,{\left (189 \, A + 136 \, C\right )} a \cos \left (d x + c\right )^{4} +{\left (189 \, A + 136 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \,{\left (21 \, A + 34 \, C\right )} a \cos \left (d x + c\right )^{2} + 85 \, C a \cos \left (d x + c\right ) + 35 \, C a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 4.73506, size = 362, normalized size = 2.08 \begin{align*} \frac{4 \,{\left (315 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 315 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (945 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 525 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (1071 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 819 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (567 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 423 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) - 2 \,{\left (63 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 47 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{315 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]